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2021 AMC Fall 10A Problem 25

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2021 AMC Fall 10A Problem 25AMC 10 Step by Step
Name:                             Date: September 13, 2026
Quadratics · difficulty 5 of 5 · about 12 min · position P21-25

A quadratic polynomial with real coefficients and leading coefficient 11 is called disrespectful\emph{disrespectful} if the equation p(p(x))=0p(p(x))=0 is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial p~(x)\tilde{p}(x) for which the sum of the roots is maximized. What is p~(1)\tilde{p}(1) ?

  1. A)

    516\frac{5}{16}

  2. B)

    12\frac{1}{2}

  3. C)

    58\frac{5}{8}

  4. D)

    11

  5. E)

    98\frac{9}{8}

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